Why is Random forcing with $\mathbb{R}$, $2^\omega$, $\omega^\omega$ all the same? By random forcing, I mean the partial order of Borel sets of the given space, modulo Lebesgue null sets, ordered by inclusion. I can not find a source proving that all these partial orders are forcing equivalent. Can anyone provide a source or mention why? Thanks.
 A: Let $X, Y$ be two of the spaces you mention. Then we can construct a Borel function $f: X\rightarrow Y$ such that for some null set $N\subset X$, $f\upharpoonright X\setminus N$ is a bijection between $X\setminus N$ and $Y$.
Now given a condition $p$ in $R_Y$ (the random forcing associated to $Y$), consider $f^{-1}(p)$. This is a condition in $R_X$ (keep in mind that Borel functions are measurable!), and embeds $R_Y$ into $R_X$ densely. 
A: The fact that the three spaces you mentioned are Borel isomorphic isn't enough. As Blass commented, the point is that the Borel isomorphism can be chosen to be null preserving which implies that the measure algebras are pairwise isomorphic. 
Random forcing refers to forcing with (an atomless) measure algebra - For example the measure algebra on the product measure space $2^{\kappa}$. Measure algebras were characterized by Maharam's theorem. The measure algebras of the standard measures on $\mathbb{R}, 2^{\omega}, \omega^{\omega}$ are each of Maharam type $\aleph_0$ and hence isomorphic as Boolean algebras.
